Inhaltsverzeichnis

Development of
Algorithmic Constructions

11:18:38
Deutsch
29.Mar 2024

Polynom = x^2+19

0. Sequence

1. Algorithm

2. Mathematical background

3. Correctness of the algorithm

4. Infinity of the sequence

5. Sequence of the polynom with 1

6. Sequence of the polynom (only primes)

7. Distribution of the primes

8. Check for existing Integer Sequences by OEIS

0. Sequence

f(0) = 19 = 19
f(1) = 5 = 5
f(2) = 23 = 23
f(3) = 7 = 7
f(4) = 35 = 5*7
f(5) = 11 = 11
f(6) = 55 = 5*11
f(7) = 17 = 17
f(8) = 83 = 83
f(9) = 25 = 5*5
f(10) = 119 = 7*17
f(11) = 35 = 5*7
f(12) = 163 = 163
f(13) = 47 = 47
f(14) = 215 = 5*43
f(15) = 61 = 61
f(16) = 275 = 5*5*11
f(17) = 77 = 7*11
f(18) = 343 = 7*7*7
f(19) = 95 = 5*19
f(20) = 419 = 419
f(21) = 115 = 5*23
f(22) = 503 = 503
f(23) = 137 = 137
f(24) = 595 = 5*7*17
f(25) = 161 = 7*23
f(26) = 695 = 5*139
f(27) = 187 = 11*17
f(28) = 803 = 11*73
f(29) = 215 = 5*43
f(30) = 919 = 919
f(31) = 245 = 5*7*7
f(32) = 1043 = 7*149
f(33) = 277 = 277
f(34) = 1175 = 5*5*47
f(35) = 311 = 311
f(36) = 1315 = 5*263
f(37) = 347 = 347
f(38) = 1463 = 7*11*19
f(39) = 385 = 5*7*11
f(40) = 1619 = 1619
f(41) = 425 = 5*5*17
f(42) = 1783 = 1783
f(43) = 467 = 467
f(44) = 1955 = 5*17*23
f(45) = 511 = 7*73
f(46) = 2135 = 5*7*61
f(47) = 557 = 557
f(48) = 2323 = 23*101
f(49) = 605 = 5*11*11
f(50) = 2519 = 11*229
f(51) = 655 = 5*131
f(52) = 2723 = 7*389
f(53) = 707 = 7*101
f(54) = 2935 = 5*587
f(55) = 761 = 761
f(56) = 3155 = 5*631
f(57) = 817 = 19*43
f(58) = 3383 = 17*199
f(59) = 875 = 5*5*5*7
f(60) = 3619 = 7*11*47
f(61) = 935 = 5*11*17
f(62) = 3863 = 3863
f(63) = 997 = 997
f(64) = 4115 = 5*823
f(65) = 1061 = 1061
f(66) = 4375 = 5*5*5*5*7
f(67) = 1127 = 7*7*23
f(68) = 4643 = 4643
f(69) = 1195 = 5*239
f(70) = 4919 = 4919
f(71) = 1265 = 5*11*23
f(72) = 5203 = 11*11*43
f(73) = 1337 = 7*191
f(74) = 5495 = 5*7*157
f(75) = 1411 = 17*83
f(76) = 5795 = 5*19*61
f(77) = 1487 = 1487
f(78) = 6103 = 17*359
f(79) = 1565 = 5*313
f(80) = 6419 = 7*7*131
f(81) = 1645 = 5*7*47
f(82) = 6743 = 11*613
f(83) = 1727 = 11*157
f(84) = 7075 = 5*5*283
f(85) = 1811 = 1811
f(86) = 7415 = 5*1483
f(87) = 1897 = 7*271
f(88) = 7763 = 7*1109
f(89) = 1985 = 5*397
f(90) = 8119 = 23*353
f(91) = 2075 = 5*5*83
f(92) = 8483 = 17*499
f(93) = 2167 = 11*197
f(94) = 8855 = 5*7*11*23
f(95) = 2261 = 7*17*19
f(96) = 9235 = 5*1847
f(97) = 2357 = 2357
f(98) = 9623 = 9623
f(99) = 2455 = 5*491
f(100) = 10019 = 43*233

1. Algorithm

If you are interested in some better algorithms have a look at quadr_Sieb_x^2+1.php.

2. Mathematical background

Lemma: If p | f(x) then also p | f(x+p) and p | f(-x-b/a) a) p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(x+p) <=> a(x+p)^2 + b(x+p) + c = 0 mod p <=> ax^2 + 2axp + ap^2 + bx + bp + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(x+p) b) if b = 0 mod a p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(-x-b/a) <=> a(-x-b/a)^2 + b(-x-b/a) + c = 0 mod p <=> ax^2 + 2bx + b^2/a - bx - b^2/a + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(-x-b/a)

3. Correctness of the algorithm

The proof for this polynom is similar to the proof for the polynom f(x)=x^2-4x+1. a) First terms for the polynom f(x) = x^2+19

f(0)=19
f(1)=5
f(2)=23
f(3)=7
f(4)=1
f(5)=11
f(6)=1
f(7)=17
f(8)=83
f(9)=1
f(10)=1
f(11)=1
f(12)=163
f(13)=47
f(14)=43
f(15)=61
f(16)=1
f(17)=1
f(18)=1
f(19)=1
f(20)=419
f(21)=1
f(22)=503
f(23)=137
f(24)=1
f(25)=1
f(26)=139
f(27)=1
f(28)=73
f(29)=1
f(30)=919
f(31)=1
f(32)=149
f(33)=277
f(34)=1
f(35)=311
f(36)=263
f(37)=347
f(38)=1
f(39)=1
f(40)=1619
f(41)=1
f(42)=1783
f(43)=467
f(44)=1
f(45)=1
f(46)=1
f(47)=557
f(48)=101
f(49)=1
f(50)=229
f(51)=131
f(52)=389
f(53)=1
f(54)=587
f(55)=761
f(56)=631
f(57)=1
f(58)=199
f(59)=1
f(60)=1
f(61)=1
f(62)=3863
f(63)=997
f(64)=823
f(65)=1061
f(66)=1
f(67)=1
f(68)=4643
f(69)=239
f(70)=4919
f(71)=1
f(72)=1
f(73)=191
f(74)=157
f(75)=1
f(76)=1
f(77)=1487
f(78)=359
f(79)=313
f(80)=1
f(81)=1
f(82)=613
f(83)=1
f(84)=283
f(85)=1811
f(86)=1483
f(87)=271
f(88)=1109
f(89)=397
f(90)=353
f(91)=1
f(92)=499
f(93)=197
f(94)=1
f(95)=1
f(96)=1847
f(97)=2357
f(98)=9623
f(99)=491

b) Substitution of the polynom
The polynom f(x)=x^2+19 could be written as f(y)= y^2+19 with x=y+0

c) Backsubstitution Beside by backsubstitution you get an estimation for the huge of the primes with p | f(x) and p < f(x) f'(y)>(2y-1) with with y=x+0
f'(x)>2x-1 with x > 4

4. Infinity of the sequence

The mathematical proof is analogue to the proof for the polynom f(x)=x^2+1

5. Sequence of the polynom with 1

19, 5, 23, 7, 1, 11, 1, 17, 83, 1, 1, 1, 163, 47, 43, 61, 1, 1, 1, 1, 419, 1, 503, 137, 1, 1, 139, 1, 73, 1, 919, 1, 149, 277, 1, 311, 263, 347, 1, 1, 1619, 1, 1783, 467, 1, 1, 1, 557, 101, 1, 229, 131, 389, 1, 587, 761, 631, 1, 199, 1, 1, 1, 3863, 997, 823, 1061, 1, 1, 4643, 239, 4919, 1, 1, 191, 157, 1, 1, 1487, 359, 313, 1, 1, 613, 1, 283, 1811, 1483, 271, 1109, 397, 353, 1, 499, 197, 1, 1, 1847, 2357, 9623, 491, 233, 1, 1489, 2657, 1, 251, 2251, 1, 1669, 1, 12119, 617, 739, 1, 1, 1, 1, 1, 1, 709, 14419, 733, 2129, 541, 3079, 3911, 1, 367, 349, 1, 2417, 859, 17443, 1, 719, 4561, 1, 1, 1733, 967, 853, 1, 20183, 1, 593, 5261, 1, 5407, 1993, 1, 3217, 1, 1217, 5857, 1, 6011, 4871, 881, 1, 1, 1, 1297, 26263, 1, 769, 1, 1103, 6977, 463, 1429, 1, 1, 4229, 7487, 1, 1, 6199, 461, 647, 1, 1907, 1, 1, 8377, 1, 1223, 1, 8747, 35363, 1787, 1901, 1, 479, 1, 443, 9511, 7687, 571, 643, 1, 5717, 1, 40823, 937, 757, 457, 1213, 1531, 43283, 1, 44119, 1, 44963, 1621, 1, 1051, 1867, 11777, 47543, 2399, 6917, 1, 1049, 12437, 10039, 1151, 929, 1, 1, 1, 52919, 1, 2341, 13577, 1, 1973, 1013, 1277, 56663, 2857, 1, 1, 8369, 14767, 1, 883, 12107, 1, 1, 1, 619, 1, 1, 16007, 12907, 1, 1873, 1, 6053, 1, 67619, 3407, 577, 1, 1, 1033, 1, 17827, 71843, 1, 947, 3673, 1721, 18637, 653, 18911, 1, 2741, 3361, 1, 7129, 1, 4679, 2861, 1, 1069, 16363, 1, 82963, 4177, 1, 1, 7753, 21467, 17291, 1, 1031, 1, 12689, 1, 90019, 1, 8293, 2087, 1, 3323, 18731, 23567, 1, 1, 1, 691, 1987, 1, 1, 1, 1, 25127, 14449, 727, 1, 5153, 103703, 1373, 1, 1, 3037, 26737, 107603, 5413, 1, 5479, 15749, 1, 4463, 2551, 2053, 1, 701, 821, 1, 1163, 1, 1279, 23671, 29761, 1, 1, 121123, 6091, 7207, 1, 123923, 4451, 3581, 31511, 1, 2897, 1, 1289, 18517, 1, 131063, 1, 1559, 33311, 1, 1, 1759, 1, 5953, 6883, 138403, 809, 1, 5023, 28279, 35537, 142903, 1, 1, 1, 20849, 36677, 1, 37061, 29803, 37447, 1, 1, 13829, 1, 1, 1, 31051, 5573, 4481, 1, 9319, 1, 160019, 1, 2099, 5801, 32647, 41011, 32971, 1, 1, 1, 1, 8447, 1, 3877, 34283, 1, 1, 6211, 1, 8779, 176419, 8863, 178103, 1, 1, 45161, 36299, 45587, 183203, 9203, 26417, 1327, 10979, 46877, 1, 1, 1, 1, 27409, 1, 193619, 1, 3203, 1, 1, 1, 3617, 49957, 200723, 1, 202519, 1453, 1, 51307, 41227, 1, 1, 1, 1303, 1, 211619, 10627, 9281, 53597, 43063, 7723, 1, 4957, 19913, 1, 220919, 11093, 4547, 1, 44939, 2969, 45319, 1, 20773, 1, 32917, 1, 1, 1, 9371, 2557, 1, 1, 238163, 1087, 1, 2411, 242083, 8681, 1, 61261, 49207, 61757, 1259, 12451, 1, 1, 5861, 1, 1, 63761, 1, 9181, 1, 2591, 1, 1187, 23833, 3463, 7549, 9473, 10651, 3931, 268343, 13469, 15907, 1, 3539, 6217, 54919, 1, 55339, 3019, 39829, 1999, 1, 1, 14897, 1, 1, 10223, 8209, 4241, 289463, 1, 291619, 2927, 41969, 10531, 5381, 1, 59627, 1, 300323, 2153, 1879, 1, 304723, 1, 1, 7001, 1, 1583, 3083, 1, 313619, 15737, 1201, 11321, 1, 79811, 1, 7307, 18979, 16189, 1, 1, 327203, 1, 65899, 1, 66359, 1, 4339, 16763, 336419, 16879, 2437, 84977, 1949, 1, 68683, 1, 1, 1, 1, 1, 50069, 87917, 70571, 1451, 3089, 89107, 1, 1, 1423, 18061, 21319, 90907, 1, 1, 1499, 1, 19457, 3709, 33829, 1697, 1, 13421, 75403, 94561, 1, 95177, 381943, 1, 54917, 1753, 2069, 4219, 1657, 1601, 11197, 1, 2647, 1, 396919, 1, 36313, 1301, 2297, 100811, 4759, 1, 1, 1, 1, 1, 17921, 9397, 7541, 2213, 1, 14951, 1, 21061, 1493, 21191, 425123, 6271, 1, 1, 1, 1, 432983, 1, 435619, 3121, 1, 109897, 1, 1, 1613, 111227, 1, 1, 26407, 1, 5441, 6661, 2113, 16273, 1, 1, 459703, 23053, 462419, 23189, 66449, 16661, 1, 1607, 1543, 1, 1871, 3391, 4001, 1, 1, 120067, 96331, 1, 13841, 17351, 44293, 2221, 490019, 24571, 1, 1, 1, 1, 1, 7351, 3659, 1, 6547, 1, 506963, 1, 101963, 5557, 20507, 1, 1567, 25849, 47129, 1, 27437, 130687, 1, 18773, 1, 132137, 1, 26573, 532919, 1, 6959, 134327, 1, 1, 108343, 1, 1, 1, 547619, 1, 50053, 12547, 110711, 1, 15901, 2287, 559523, 28051, 562519, 28201, 80789, 1, 10337, 1, 114311, 6229, 1, 1, 1, 1, 580663, 145547, 10613, 1, 1, 21011, 1, 29569, 592919, 29723, 35059, 21341, 17117, 1, 10949, 1, 9923, 1597, 3779, 4357, 611543, 153277, 1, 154061, 1, 2011, 2063, 1831, 624119, 6257, 36899, 157217, 18013, 22573, 1, 14437, 1, 1, 640019, 4583, 13127, 1, 129287, 162011, 7643, 7079, 1, 1, 656119, 32887, 14029, 165247, 132523, 3389, 1, 8783, 1, 3049, 61129, 33703, 1, 1, 135799, 1, 1, 1, 1, 4909, 1, 1, 2971, 3691, 27823, 174311, 1, 1, 8461, 1, 41507, 1, 64453, 1, 20353, 1, 3329, 179357, 9851, 1, 103217, 1, 65993, 1, 1, 9619, 1, 1, 1, 1, 1, 1, 4987, 16927, 1, 26723, 1, 187927, 4799, 1, 756919, 5419, 4723, 190537, 1, 17401, 6673, 11311, 110129, 5519, 16477, 1, 4073, 1, 31259, 2543, 2039, 1, 788563, 1, 792119, 1, 1, 1, 9403, 8707, 1, 18287, 2593, 1, 1, 40591, 813623, 203857, 2239, 2467, 1, 2671, 4409, 8263, 828119, 2441, 43777, 4253, 23869, 209311, 33563, 4889, 3331, 1, 120917, 1, 1, 212987, 170759, 12583, 1, 1, 7237, 3923, 1, 2281, 20201, 1, 1, 31223, 1, 1, 879863, 44087, 80329, 1, 7457, 222317, 178231, 1, 178987, 1, 128389, 1, 47501, 4111, 82393, 4831, 182027, 32573, 26113, 2267, 917783, 1, 21433, 1, 1, 3011, 8081, 232811, 1, 233777, 1, 1, 134417, 47143, 944803, 21517, 1, 5527, 1, 1, 956503, 2819, 960419, 48119, 7039, 34511, 1, 22051, 194443, 10589, 1, 48907, 1, 1, 984083, 1, 2707, 22501, 1, 1, 20327, 1,

6. Sequence of the polynom (only primes)

19, 5, 23, 7, 11, 17, 83, 163, 47, 43, 61, 419, 503, 137, 139, 73, 919, 149, 277, 311, 263, 347, 1619, 1783, 467, 557, 101, 229, 131, 389, 587, 761, 631, 199, 3863, 997, 823, 1061, 4643, 239, 4919, 191, 157, 1487, 359, 313, 613, 283, 1811, 1483, 271, 1109, 397, 353, 499, 197, 1847, 2357, 9623, 491, 233, 1489, 2657, 251, 2251, 1669, 12119, 617, 739, 709, 14419, 733, 2129, 541, 3079, 3911, 367, 349, 2417, 859, 17443, 719, 4561, 1733, 967, 853, 20183, 593, 5261, 5407, 1993, 3217, 1217, 5857, 6011, 4871, 881, 1297, 26263, 769, 1103, 6977, 463, 1429, 4229, 7487, 6199, 461, 647, 1907, 8377, 1223, 8747, 35363, 1787, 1901, 479, 443, 9511, 7687, 571, 643, 5717, 40823, 937, 757, 457, 1213, 1531, 43283, 44119, 44963, 1621, 1051, 1867, 11777, 47543, 2399, 6917, 1049, 12437, 10039, 1151, 929, 52919, 2341, 13577, 1973, 1013, 1277, 56663, 2857, 8369, 14767, 883, 12107, 619, 16007, 12907, 1873, 6053, 67619, 3407, 577, 1033, 17827, 71843, 947, 3673, 1721, 18637, 653, 18911, 2741, 3361, 7129, 4679, 2861, 1069, 16363, 82963, 4177, 7753, 21467, 17291, 1031, 12689, 90019, 8293, 2087, 3323, 18731, 23567, 691, 1987, 25127, 14449, 727, 5153, 103703, 1373, 3037, 26737, 107603, 5413, 5479, 15749, 4463, 2551, 2053, 701, 821, 1163, 1279, 23671, 29761, 121123, 6091, 7207, 123923, 4451, 3581, 31511, 2897, 1289, 18517, 131063, 1559, 33311, 1759, 5953, 6883, 138403, 809, 5023, 28279, 35537, 142903, 20849, 36677, 37061, 29803, 37447, 13829, 31051, 5573, 4481, 9319, 160019, 2099, 5801, 32647, 41011, 32971, 8447, 3877, 34283, 6211, 8779, 176419, 8863, 178103, 45161, 36299, 45587, 183203, 9203, 26417, 1327, 10979, 46877, 27409, 193619, 3203, 3617, 49957, 200723, 202519, 1453, 51307, 41227, 1303, 211619, 10627, 9281, 53597, 43063, 7723, 4957, 19913, 220919, 11093, 4547, 44939, 2969, 45319, 20773, 32917, 9371, 2557, 238163, 1087, 2411, 242083, 8681, 61261, 49207, 61757, 1259, 12451, 5861, 63761, 9181, 2591, 1187, 23833, 3463, 7549, 9473, 10651, 3931, 268343, 13469, 15907, 3539, 6217, 54919, 55339, 3019, 39829, 1999, 14897, 10223, 8209, 4241, 289463, 291619, 2927, 41969, 10531, 5381, 59627, 300323, 2153, 1879, 304723, 7001, 1583, 3083, 313619, 15737, 1201, 11321, 79811, 7307, 18979, 16189, 327203, 65899, 66359, 4339, 16763, 336419, 16879, 2437, 84977, 1949, 68683, 50069, 87917, 70571, 1451, 3089, 89107, 1423, 18061, 21319, 90907, 1499, 19457, 3709, 33829, 1697, 13421, 75403, 94561, 95177, 381943, 54917, 1753, 2069, 4219, 1657, 1601, 11197, 2647, 396919, 36313, 1301, 2297, 100811, 4759, 17921, 9397, 7541, 2213, 14951, 21061, 1493, 21191, 425123, 6271, 432983, 435619, 3121, 109897, 1613, 111227, 26407, 5441, 6661, 2113, 16273, 459703, 23053, 462419, 23189, 66449, 16661, 1607, 1543, 1871, 3391, 4001, 120067, 96331, 13841, 17351, 44293, 2221, 490019, 24571, 7351, 3659, 6547, 506963, 101963, 5557, 20507, 1567, 25849, 47129, 27437, 130687, 18773, 132137, 26573, 532919, 6959, 134327, 108343, 547619, 50053, 12547, 110711, 15901, 2287, 559523, 28051, 562519, 28201, 80789, 10337, 114311, 6229, 580663, 145547, 10613, 21011, 29569, 592919, 29723, 35059, 21341, 17117, 10949, 9923, 1597, 3779, 4357, 611543, 153277, 154061, 2011, 2063, 1831, 624119, 6257, 36899, 157217, 18013, 22573, 14437, 640019, 4583, 13127, 129287, 162011, 7643, 7079, 656119, 32887, 14029, 165247, 132523, 3389, 8783, 3049, 61129, 33703, 135799, 4909, 2971, 3691, 27823, 174311, 8461, 41507, 64453, 20353, 3329, 179357, 9851, 103217, 65993, 9619, 4987, 16927, 26723, 187927, 4799, 756919, 5419, 4723, 190537, 17401, 6673, 11311, 110129, 5519, 16477, 4073, 31259, 2543, 2039, 788563, 792119, 9403, 8707, 18287, 2593, 40591, 813623, 203857, 2239, 2467, 2671, 4409, 8263, 828119, 2441, 43777, 4253, 23869, 209311, 33563, 4889, 3331, 120917, 212987, 170759, 12583, 7237, 3923, 2281, 20201, 31223, 879863, 44087, 80329, 7457, 222317, 178231, 178987, 128389, 47501, 4111, 82393, 4831, 182027, 32573, 26113, 2267, 917783, 21433, 3011, 8081, 232811, 233777, 134417, 47143, 944803, 21517, 5527, 956503, 2819, 960419, 48119, 7039, 34511, 22051, 194443, 10589, 48907, 984083, 2707, 22501, 20327,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x+19 and
the reducible primes which appear as divisor for the first time
p | x^2x+19 and p < x^2x+19

To avoid confusion with the number of primes:
I did not count the primes <= A
but I counted the primes appending the x and therefore the x <= A

ABCDEFGHIJK
exponent =log10 (x)<=xnumber of all primesnumber of primes p = f(x) number of primes p | f(x) C/xD/xE/xC(n) / C(n-1)D(n) / D(n-1)E(n) / E(n-1)
1107700.7000000.7000000.7000000.0000000.0000000.000000
21006131300.6100000.3100000.6100008.7142864.428571inf
31.0006291754540.6290000.1750000.62900010.3114765.64516115.133333
410.0006.4971.2555.2420.6497000.1255000.64970010.3290947.17142911.546255
5100.00066.0289.38856.6400.6602800.0938800.66028010.1628457.48047810.805037
61.000.000666.11376.277589.8360.6661130.0762770.66611310.0883428.12494710.413772
710.000.0006.699.263641.4336.057.8300.6699260.0641430.66992610.0572478.40925810.270363
8100.000.00067.276.8035.544.36061.732.4430.6727680.0554440.67276810.0424188.64370810.190521
91.000.000.000675.003.13848.786.087626.217.0510.6750030.0487860.67500310.0332238.79922810.144051
1010.000.000.0006.767.832.235435.760.7706.332.071.4650.6767830.0435760.67678310.0263718.93207110.111625


ABCDEFGHIJK
exponent =log2 (x)<=xnumber of all primesnumber of primes p = f(x) number of primes p | f(x) C/xD/xE/xC(n) / C(n-1)D(n) / D(n-1)E(n) / E(n-1)
123301.5000001.5000000.0000000.0000000.0000000.000000
244401.0000001.0000000.0000001.3333331.333333-nan
387700.8750000.8750000.0000001.7500001.750000-nan
416111010.6875000.6250000.0625001.5714291.428571inf
532181440.5625000.4375000.1250001.6363641.4000004.000000
6643724130.5781250.3750000.2031252.0555561.7142863.250000
71287835430.6093750.2734380.3359382.1081081.4583333.307692
825616061990.6250000.2382810.3867192.0512821.7428572.302325
95123161072090.6171880.2089840.4082031.9750001.7540982.111111
101.0246461794670.6308590.1748050.4560552.0443041.6728972.234450
112.0481.3083219870.6386720.1567380.4819342.0247681.7932962.113490
124.0962.6345852.0490.6430660.1428220.5002442.0137621.8224302.075988
138.1925.3141.0654.2490.6486820.1300050.5186772.0174641.8205132.073694
1416.38410.7251.9238.8020.6546020.1173710.5372312.0182541.8056342.071546
1532.76821.5173.52217.9950.6566470.1074830.5491642.0062471.8315132.044422
1665.53643.1976.41336.7840.6591340.0978550.5612792.0075761.8208402.044123
17131.07286.70911.99674.7130.6615370.0915220.5700152.0072921.8705752.031128
18262.144173.86322.450151.4130.6632350.0856400.5775952.0051321.8714572.026595
19524.288348.65242.157306.4950.6650010.0804080.5845932.0053261.8778172.024232
201.048.576698.53079.678618.8520.6661700.0759870.5901832.0035161.8900302.019126
212.097.1521.399.841150.9031.248.9380.6674960.0719560.5955402.0039811.8939112.018153
224.194.3042.804.419286.2052.518.2140.6686260.0682370.6003892.0033841.8966162.016284
238.388.6085.617.943544.5565.073.3870.6697110.0649160.6047952.0032471.9026782.014677
2416.777.21611.251.3341.039.02310.212.3110.6706320.0619310.6087012.0027501.9080192.012918
2533.554.43222.532.6491.988.27120.544.3780.6715250.0592550.6122702.0026651.9135972.011727
2667.108.86445.118.4583.810.63841.307.8200.6723170.0567830.6155342.0023591.9165592.010663
27134.217.72890.339.1977.314.95683.024.2410.6730790.0545010.6185792.0022671.9196152.009892
28268.435.456180.869.87114.061.194166.808.6770.6737930.0523820.6214112.0021201.9222532.009156
29536.870.912362.090.84627.069.406335.021.4400.6744470.0504210.6240262.0019411.9251142.008417
301.073.741.824724.845.37852.189.035672.656.3430.6750650.0486050.6264602.0018331.9279712.007801
312.147.483.6481.450.921.648100.765.3871.350.156.2610.6756380.0469230.6287162.0016981.9307772.007201
324.294.967.2962.904.143.862194.788.6402.709.355.2220.6761740.0453530.6308212.0015861.9330912.006697
338.589.934.5925.812.616.475376.968.2335.435.648.2420.6766780.0438850.6327932.0014911.9352682.006252
3417.179.869.18411.633.399.706730.297.06210.903.102.6440.6771530.0425090.6346442.0014051.9372912.005852
3534.359.738.36823.282.309.7271.416.203.34121.866.106.3860.6776040.0412170.6363872.0013331.9392152.005494
3668.719.476.73646.593.812.3552.748.862.99543.844.949.3600.6780290.0400010.6380282.0012541.9410092.005156


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p=f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
123120111
244220112
387251312
41610461423
53214592525
664248163858
712835926411713
82566118431017925
9512107357215381737
101.0241796311625682363
112.0483211112104412142114
124.0965851953908722174203
138.1921.065357708154399134378
1416.3841.9236451.278247716258702
1532.7683.5221.1832.3394611.2974691.295
1665.5366.4132.1764.2378472.3638322.371
17131.07211.9964.0537.9431.5694.4551.5474.425
18262.14422.4507.50114.9492.9348.3272.8888.301
19524.28842.15714.05028.1075.49215.6925.39015.583
201.048.57679.67826.51953.15910.29529.70210.14329.538
212.097.152150.90350.245100.65819.44656.28719.30655.864
224.194.304286.20595.357190.84836.684106.66436.562106.295
238.388.608544.556181.245363.31169.583202.56869.720202.685
2416.777.2161.039.023345.899693.124132.933386.523133.109386.458
2533.554.4321.988.271662.5671.325.704253.907740.083254.230740.051
2667.108.8643.810.6381.270.4922.540.146486.4641.419.390486.3171.418.467
27134.217.7287.314.9562.439.0324.875.924932.6852.725.750931.8032.724.718
28268.435.45614.061.1944.688.6129.372.5821.790.9655.240.5531.790.5565.239.120
29536.870.91227.069.4069.024.77618.044.6303.445.39510.089.8733.445.10210.089.036
301.073.741.82452.189.03517.398.03434.791.0016.640.29719.455.6216.638.54919.454.568
312.147.483.648100.765.38733.589.37567.176.01212.813.11937.574.28912.810.57137.567.408
324.294.967.296194.788.64064.937.069129.851.57124.756.90772.644.00024.750.82772.636.906
338.589.934.592376.968.233125.669.788251.298.44547.889.915140.600.24047.879.898140.598.180
3417.179.869.184730.297.062243.449.532486.847.53092.716.404272.446.53692.712.172272.421.950
3534.359.738.3681.416.203.341472.083.254944.120.087179.710.039528.417.769179.699.888528.375.645
3668.719.476.7362.748.862.995916.295.5211.832.567.474348.651.1171.025.775.499348.662.0041.025.774.375


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p|f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
120000000
240000000
380000000
4161100100
5324311210
66413761444
71284325187111411
825699534622242924
951220910910048565550
101.024467244223111120121115
112.048987516471252243257235
124.0962.0491.078971525513511500
138.1924.2492.2072.0421.1051.0611.0641.019
1416.3848.8024.5284.2742.2622.1892.2302.121
1532.76817.9959.2718.7244.6004.4174.5414.437
1665.53636.78418.95717.8279.3199.0399.2739.153
17131.07274.71338.42136.29218.85518.47318.85618.529
18262.144151.41377.34074.07338.11937.45938.40637.429
19524.288306.495156.400150.09577.45475.80977.84375.389
201.048.576618.852315.687303.165156.505152.862156.637152.848
212.097.1521.248.938636.385612.553315.676309.070315.538308.654
224.194.3042.518.2141.282.2191.235.995635.583623.577635.584623.470
238.388.6085.073.3872.582.6602.490.7271.279.3771.256.7881.279.7121.257.510
2416.777.21610.212.3115.192.9215.019.3902.574.7822.531.4732.574.0222.532.034
2533.554.43220.544.37810.436.63310.107.7455.176.1985.095.0175.176.8895.096.274
2667.108.86441.307.82020.970.07920.337.74110.402.98710.249.34410.402.39810.253.091
27134.217.72883.024.24142.112.31340.911.92820.899.43220.612.11120.903.22720.609.471
28268.435.456166.808.67784.562.88782.245.79041.977.91841.429.70941.979.10441.421.946
29536.870.912335.021.440169.725.023165.296.41784.278.51183.230.55984.290.47683.221.894
301.073.741.824672.656.343340.598.908332.057.435169.174.500167.157.920169.188.080167.135.843
312.147.483.6481.350.156.261683.319.079666.837.182339.476.078335.601.512339.507.578335.571.093
324.294.967.2962.709.355.2221.370.589.2801.338.765.942681.097.587673.595.131681.126.806673.535.698
338.589.934.5925.435.648.2422.748.593.5852.687.054.6571.366.163.6361.351.675.4461.366.189.7511.351.619.409
3417.179.869.18410.903.102.6445.511.139.3855.391.963.2592.739.764.9112.711.772.5912.739.810.2622.711.754.880
3534.359.738.36821.866.106.38611.048.434.38510.817.672.0015.493.633.8165.439.387.3365.493.695.2945.439.389.940
3668.719.476.73643.844.949.36022.146.120.79321.698.828.56711.013.686.77110.908.719.23511.013.824.12310.908.719.231


8. Check for existing Integer Sequences by OEIS

Found in Database : 19, 5, 23, 7, 1, 11, 1, 17, 83, 1, 1, 1, 163, 47, 43, 61, 1, 1, 1, 1,
Found in Database : 19, 5, 23, 7, 11, 17, 83, 163, 47, 43, 61, 419, 503, 137, 139, 73, 919, 149, 277, 311, 263, 347,
Found in Database : 5, 7, 11, 17, 19, 23, 43, 47, 61, 73, 83, 101, 131, 137, 139, 149,